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Mirrors > Home > ILE Home > Th. List > caofrss | Unicode version |
Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | |
caofref.2 | |
caofcom.3 | |
caofrss.4 |
Ref | Expression |
---|---|
caofrss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.2 | . . . . 5 | |
2 | 1 | ffvelrnda 5302 | . . . 4 |
3 | caofcom.3 | . . . . 5 | |
4 | 3 | ffvelrnda 5302 | . . . 4 |
5 | caofrss.4 | . . . . . 6 | |
6 | 5 | ralrimivva 2401 | . . . . 5 |
7 | 6 | adantr 261 | . . . 4 |
8 | breq1 3767 | . . . . . 6 | |
9 | breq1 3767 | . . . . . 6 | |
10 | 8, 9 | imbi12d 223 | . . . . 5 |
11 | breq2 3768 | . . . . . 6 | |
12 | breq2 3768 | . . . . . 6 | |
13 | 11, 12 | imbi12d 223 | . . . . 5 |
14 | 10, 13 | rspc2va 2663 | . . . 4 |
15 | 2, 4, 7, 14 | syl21anc 1134 | . . 3 |
16 | 15 | ralimdva 2387 | . 2 |
17 | ffn 5046 | . . . 4 | |
18 | 1, 17 | syl 14 | . . 3 |
19 | ffn 5046 | . . . 4 | |
20 | 3, 19 | syl 14 | . . 3 |
21 | caofref.1 | . . 3 | |
22 | inidm 3146 | . . 3 | |
23 | eqidd 2041 | . . 3 | |
24 | eqidd 2041 | . . 3 | |
25 | 18, 20, 21, 21, 22, 23, 24 | ofrfval 5720 | . 2 |
26 | 18, 20, 21, 21, 22, 23, 24 | ofrfval 5720 | . 2 |
27 | 16, 25, 26 | 3imtr4d 192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 wral 2306 class class class wbr 3764 wfn 4897 wf 4898 cfv 4902 cofr 5711 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ofr 5713 |
This theorem is referenced by: (None) |
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